We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem where is the vector fieldwith a boundedness condition on the divergence of each vector field . This model was studied in the paper [LL] with a regularity assumption replacing our hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of [Am] and we combine some consequences of the Alberti rank-one structure theorem for vector fields with a regularization procedure. Our regularization kernel is not restricted to be a convolution and is introduced as an unknown function. Our method amounts to commute a pseudo-differential operator with a function.
@article{ASNSP_2004_5_3_4_681_0, author = {Lerner, Nicolas}, title = {Transport equations with partially $BV$ velocities}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {681--703}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 3}, number = {4}, year = {2004}, mrnumber = {2124585}, zbl = {1170.35362}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2004_5_3_4_681_0/} }
TY - JOUR AU - Lerner, Nicolas TI - Transport equations with partially $BV$ velocities JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2004 SP - 681 EP - 703 VL - 3 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2004_5_3_4_681_0/ LA - en ID - ASNSP_2004_5_3_4_681_0 ER -
%0 Journal Article %A Lerner, Nicolas %T Transport equations with partially $BV$ velocities %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2004 %P 681-703 %V 3 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2004_5_3_4_681_0/ %G en %F ASNSP_2004_5_3_4_681_0
Lerner, Nicolas. Transport equations with partially $BV$ velocities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 4, pp. 681-703. http://www.numdam.org/item/ASNSP_2004_5_3_4_681_0/
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